68 REPORT — 1856. 



gas burned, we have determined by accurate experiment, so that between 

 certain limits we can calculate the amount of action produced by burning 

 the unit volume of gas issuint; at a given rate. We call the unit amount of 

 action for any instrument that produced by burning a cubic millimetre of 

 carbonic oxide at the distance of one millimetre from the sensitive gas, issuing 

 under the above-mentioned circumstances. 



The interesting relations of the reflexion, absorption, and polarization of 

 the chemical rays, we hope to have the honour of laying before the Section 

 on a future occasion. 



Heidelberg, August 5th, 1856. 



On the Trigonometry of the Parabola, and the Geometrical Origin of 

 Logarithms. By the Rev. James Booth, LL.D., F.R.S. &)C. 



[A Communication ordered to be printed among the Reports.] 



When engaged, some years ago, in researches on the geometrical properties 

 of elliptic integrals, the results of which appeared in two memoirs printed in 

 the Philosophical Transactions for 1852 and 1854, 1 was led to discuss a par- 

 ticular case of a cardinal theorem in the theory of elliptic integrals. Cer- 

 tainly no discovery was anticipated in matters so long known and thoroughly 

 investigated as the theory of logarithms and the properties of the parabola. 

 The propositions I now bring before the Section are, I believe, entirely new ; 

 and as they open a field of research in a department of geometrical science 

 studied by every mathematician in the course of his reading, I thought the 

 discussion of them might not prove unacceptable to the Mathematical Section 

 of the British Association. 



Section I. 



I. Let the angles w, tp, and x^ which we shall call conjugate amplitudes, be 

 connected by the equation 



tan w=tan0secx+ tanxsec^ (1) 



Hence to is such a function of ^ and x as will render 



tan l_(j), x] = t^'" sec x+ tan x sec (j>. 

 We must adopt some appropriate notation to represent this function. Let 

 the function [^, xl be written (^-"-X) so that 



tan ((p-'- x)= tan ^secx+ tan xsec^. 



This equation must be taken as the definition of the function ^-^X' 

 In like manner we may represent by tan (<{>-rx) the expression 



tan (j) sec x" tan x sec f. 

 From (1) we obtain 



sec w=sec(0-^x)=sec 0secx4- tan^tanx (2) 



If we now differentiate the equation 



tan 0) = tan ^ sec x + tan x sec 0, 

 we shall have 



— — . sec w= ^ ■ . sec (p sec yH — tan 6 tan x 



cos W cos m cos V 1 



r*' • • (2) 



T fon A. fan *, _1_ _ A,_ 



-)- "'*" ^ . tan tan x H — -^ sec ^ sec x I 



cos'(j) cos X 



