adjusted at the other, care being taken that the micrometer be set to zero. 2. Point the telescope to the upper line of the staff, clamp it, and read on the scale the number of divisions shown by the cross lines of the microscope. Suppose it to be 67 and something more, this is therefore more than 670 000 parts of the whole scale of one million parts; the micrometer disc is then turned either way until the next division of the scale is brought exactly between the cross lines that indicate the centre of the microscope, when the vernier of the micrometer may give an additional reading of 2035; the total reading thus becomes 672035; but in case of the microscopic reading having been more than 67, the total reading would be 677 035, as the micrometer vernier subdivides. only the half of one of the numbered divisions of the horizontal scale. 3. Unclamp the telescope and the microscope, and observe similarly on the lower marked line of the staff, but in the same vertical plane, obtaining a second reading which we will suppose is 660 015. 4. Notice also the reading for the point of departure on the scale when the telescope is levelled; suppose it to be 500 010. The mode of calculation. From these three readings all the necessary results for distance, altitude, and level are obtained by the before-mentioned formulæ or proportion; where if S= 10 feet; 6 inches = 1 500 000 parts of the scale, and 1=672035-660015=12020 parts, then the distance hS D= 1 = I 500 000 X IO 12 020 =1247'92 feet. And to obtain the altitude of the lower point on the staff, here we have D given, and the value of in this case being the difference of readings between that of the lower point and the point of departure=160 005, we obtain ID ___ 160 005 × 1247'92 h I 500 000 = =133'115 feet. Should the reading for the lower point on the staff be less than that of the point of departure, the altitude thus obtained will be negative. The value of s may also be obtained by another proportion, thus, To avoid dividing the constant dividend, a small table may be made from which the distances may be obtained by inspection. Other points of importance. The following shows the amount of exactitude that may be obtained with this instrument. Distances of 100 feet exact to o'001 foot. Heights at a distance of 300 feet exact to o'0005 foot It is easily seen that the divisions of the scale may be of any convenient corresponding length, as they are merely required to represent ratios; and that the base scale may be of any length, instead of exactly four inches, so long as it is divided into 200 equal parts, and that the micrometer vernier divides each of these into 5000 equal parts; hence the instrument admits of modification without inconvenience. To determine the value (), or as it is called the base line of the instrument, measured from the point of rotation of the microscope perpendicular to the scale; measure very accurately a distance, say 100 feet, from the instrument to the 10-feet staff, and take the readings giving / as a difference, hence obtain 1 x 100 But before thus obtaining this value, it must be made sure that the optical axis of the microscope and telescope are in the same vertical plane, or rather in parallel vertical planes, and perpendicular to each other: to effect this adjustment, first sight a 10-feet staff, and next a shorter one, say of 5 feet, at the same distance, and try with the acquired data whether the base line of the instrument remains in each case proportional to the distance or not; should this not be the case, the cross-lines of the microscope must be moved by the motion of their adjusting screw. FIGURE 4.-INSTRUMENTAL CORRECTION. To determine mathematically the height of the base line of the instrument, and the position of the point of Station departure on the scale, shown as OH and RH in the figure. Let RS-a, any distance measured on the scale, subtending the known angle A; and ST=b, any convenient distance subtending the known angle B. Put the unknown angle ORH-X, and the unknown. side OS=z. Then since a z:: sin A sin X and : zb:: sin (A+B+X): sin B a b: sin A.sin (A+B+X): sin X.sin B but sin (A+B+X)=sin (A+B) cos X + cos (A + B).sin X and OH=RO sin X; and RH=RO cos X; the values of these two required quantities can be obtained. The following is the form of Field-book or record for this work recommended by Eckhold. The numerous advantages of such an instrument require little or no explanation, and the exactitude of result being undoubted, the necessity of measuring and depending on carefully-measured base lines in large surveys is entirely obviated; levelling as well as measuring can be done at very long distances on inclined planes; and having a constant starting point for levelling there is nothing further to do with the collimation of the optical axis of the telescope. This instrument has been used in railway surveys in America and on revenue surveys in India. Micrometer Eyepieces. In the same way that the Porro principle with its complicated eyepieces has resolved itself into the simple application in practice of using extra wires in level telescopes of high power, correspondingly the Eckhold principle of using both microscope and micrometer may reduce itself in practice to the simple application of micrometers to the eyepieces of theodolites. The use of moveable micrometer-eyepieces was first publicly exemplified in survey work on the topographical survey of Abyssinia. The six-inch. transit theodolites used had eyepieces furnished with a pair of micrometers fitted into a rectangular frame, which was capable of being turned round on a collar, so as to bring the micrometer wires parallel either to the horizontal or the vertical wires of the fixed diaphragm plate; a division of the micrometer was equal to about two seconds. This construction enables the angle subtended by a pole of known length, set up either in a vertical or a horizontal position, to be measured by the micrometers; distances can thus be deduced with considerable accuracy, either in base lines or in traversing; while in astronomical observations for time and longi |