TRANSACTIONS OF THE SKCTIONS. 7 



the coordinates, suppose from the point (.r" y*) toivards the other point, 

 it will, by motion in that direction, describe and increase positive 

 distances ; but, by a motion in the contrary direction it would de- 

 scribe or increase negative distances. If we then suppose the in- 

 definite line joining (o^^y^*) and {x\/) to revolve slowly round one of 

 these points as a centre, suppose round {x''' y") the law of continuity 

 will compel us to consider a point moving the same way from the 

 fixed point as describing and increasing negative distances ; now, 

 when the revolving line has completed a semirevolution, it will again 

 pass through the second point, {x'y''); but a moving point setting 

 off from (ar"y") must, in order to reach (x'y), move along the inde- 

 finite line after its semirevolution, in precisely an opposite direction 

 to that which led to the same point in its first position, and, there- 

 fore, the same identical distance must, in this last position, be con- 

 sidered as negative, if, in the first position, we assumed it to be po- 

 sitive ; and hence the double sign to which the analytic value directs 

 our attention. 



In a similar way we can explain the double sign of the secant of 

 an arch, the opposite sign of the secant of an arch, and of its sup- 

 plement, and of the same arch increased by a semicircle. We may 

 also see the reason for the double sign of the analytic value of the 

 radius of curvature ; and thus many symbols which were formerly 

 not perceived to have any relation to position, will ajipear to have a 

 very direct and intelligible relation to it ; and thus, much that was 

 formerly arbitrary will be rendered subject to precise rules. 



On " The Mathematical Rules for constructing Compensating Pendu- 

 lums." By Professor Stevelly. 

 Accident led the author to the discovery of an error of serious 

 consequence which he had previously never suspected, in the principle 

 of calculating the dimensions of the several parts of compensating pen- 

 dulums adopted by Captain Kater, and detailed by him in the latter 

 part of the volume on Mechanics in Lardner's Cyclopaedia. Doctor 

 Templeton, of the Royal Artillery, had kindly undertaken to find a 

 meridian line at the apartments of the Museum of the Belfast Natu- 

 ral History Society. When doing so he had used a well-made eight- 

 day clock furnished with a pendulum with a deal rod, which although 

 carefully made had not been intended to compensate for changes of 

 temperature. This pendulum had gone in a room immediately under 

 a leaden roof during a very cold winter, and afterwards during a very 

 hot summer, and yet had not varied more than a very few secorids 

 from mean time, and even that variation had not taken place with 

 any considerable departure from a mean rate of gaining. Surprise 

 at this fact led Mr. Stevelly to perceive that a common deal rod pen- 

 dulum, with a large lenticular leaden bob resting on a nut, and trans- 

 fixed by the deal rod, must be to a certain extent compensating. He 

 then proceeded to calculate the exact dimensions for perfect compen- 

 sation ; but upon applying the mathematical principle upon which he 

 had made the calculation to some of the examples given by Captain 



