TRANSACTIONS OF SECTION A. 457 



To render the chronometer at the bottom of the bore susceptible only to the 

 influence of the temperature within : ram over it suitable material, as over gun- 

 powder in rock-blasting operations, so as to exclude water and air. 



Sets of apparatus may be arranged to investigate the temperatures at various 

 depths. 



Observation stations. — Bores in the deeper rocks will give results of greater 

 value respecting the general conditions of the earth within. Bores- in the higher 

 strata will give results interesting chiefly as compared with those from the deeper 

 rocks, and in connection with the phenomena of earthquakes and volcanos. Stations 

 in insular and continental, below and above sea level, and in different geological 

 systems, should be chosen.* 



14. On Sunspots and Rainfall. By C. Meldeum. Ordered by the Council 

 to be printed in extenso among the Reports, see p. 230. 



15. On Lightning Conductors. By R. Anderson. 



The following Papers were read in the Department of Mathematics : — 



1. Report of the Committee on Babbage's Analytical Engine. 

 See Reports, p. 92. 



2. Report of the Committee on Mathematical Tables, with an Explanation 

 of the Mode of Formation of the Factor Table for the Fourth Million. 



See Reports, p. 172. 



3. On a Neiv Form of Tangential Equation.^ By John Casey, LL.D., F.R.S., 

 M.R.I.A., Professor of Mathematics in the Catholic University of Ireland. 



1. The tangential equation of a curve is a relation among the co-efficients in the 

 equation of a variable line, which being fulfilled the line must be a tangent to the 

 curve. Thus, let be the origin, OX, OY the axes, and let a variable line MN, in 

 any of its positions, make an angle <fi with OX, and an intercept v on it, then the 

 equation of MN is 



x + y cot (f> — v = o. 



From this it follows that if v and (f> be given, the position of the line is fixed, and 

 also that any relation between v and <p such as 



will be the tangential equation of a curve which is the envelope of the line. 



This form is remarkable for the facility with which it can be transformed into 

 all the known forms of equation, and also for the simplicity of the formulae which 

 it gives for metric purposes, such as rectification, curvature, &c. 



"We give here an outline of the transformations, &c, of which it is capable. 



2. The tangential equation v=f(<f>) ot a curve being given to find its Cartesian 

 equation. 



* The original paper was read in extenso at the sitting of the Academy of 

 Sciences, Paris, on the 16th September. 



f This paper was published in extenso in the ' Philosophical Transactions ' for the 

 year 1878, vol. 167, part 2. 



