188 Mr Hill, On the effect of [Feb. 20, 



Since sn^iu — t'tan u, a formula involving sr^'s is equivalent to 

 a trigonometrical formula involving tangents, so that the method 

 affords a means of deriving elliptic function formulas involving sn's 

 (and therefore also trigonometrical formulae involving sines) from 

 trigonometrical formulae involving tangents. 



© © O 



In Vol. in. pp. 383 — 387 a method was given of deriving from 

 a formula 2 . II sin (a — /3) = a trigonometrical formula of the 

 form 2 . II sin (a— /3) sin (a + /3) = ; but this method of deriving one 

 sine-formula from another can be applied only when the number 

 of sines multiplied together is the same in each term of the 

 formula. But in the above method, where the original formula 

 involves sn's, this condition is not necessary, viz. we put k = 1 and 

 obtain a formula involving sn a (a — /3), &c. : we then replace 

 sn, (a — /3), &c. by k sn (a — /3) sn (a + /3), &c. and we deduce a 

 trigonometrical formula involving sines by putting k = 0, and a 

 trigonometrical formula involving tangents by putting k = 1, re- 

 placing a, /3, &c. by vi, i/3, &c. and substituting ttan (a — /3), &c. 

 for sn, (iz — i/3), &c. 



February 20, 1882. 

 PiiOFESsoit Stokes in the chair. 



The following communications were made to the Society : 



(1) On the effect of fluctuations in a variable, upon the mean 

 values of functions of that variable: with an application to the theory 

 of Glacial Epochs. By E. Hill, M.A., Tutor of St John's College. 



Let a variable x be changed continuously and uniformly from 

 a magnitude a — b to another a + b : and let f(x) be a function 

 continuous between /(a — 6) and f(a + b). Then if the mean of 

 its intermediate values be taken, this will not in general coincide 

 with f\a), the value of the function for a, the mean magnitude 

 of the variable. So, in any continuous change of a variable and 

 of a function of that variable, the mean value of the function 

 will not in general be the value of it for the mean magnitude 

 of the variable. 



Take a series of values of y, uniformly increasing in magnitude, 

 such as the ordinates of the straight line PRQ. Let the corre- 

 sponding values of any functions f(y), f 2 {y) be represented by 



