380 On a Theorem in the Philosophical Magazine for August. 



But it is very trying to have two-thirds of the magnetic power 

 temporarily suppressed by mutual proximity, and they would 

 probably soon lose permanently a portion of it by any slight 

 concussion. 



The experiments were gone over three times, and the mean 

 results taken. The magnets were placed at their closest di- 

 stance before beginning, so that further permanent loss of 

 magnetism might not take place during the experiments. I 

 believe the third place of decimals in the first table cannot be 

 trusted. 



From the foregoing table and observation following, it ap- 

 pears that power is not gained by too great an approximation 

 of plates of a compound magnet, and the thinner the plates 

 the better. 



W. Petrie. 



LV. On a Theorem given in the Philosophical Magazine for 



August. 



To the Editors of the Philosophical Magazine and Journal. 

 Gentlemen, 



THE " new theorem" which is applied in the August Num 

 ber of your Magazine to determine the development of 

 cos n 9 in descending powers of cos 0, may be at once deduced 

 from one of the fundamental formulae in finite differences, as 

 follows: — the formula referred to is 



or (-1) A" « ( = u x — niix+h + 1 g u x +z& — &c, 



where u x is any function of x, and h = A x. 



Now let u x be of m dimensions in x, and = (a—x) {b — x) 



(c—x) ... (/—#), then it is evident, that if n > m t A 71 u x = 

 .-. = u x - n u x + h + ^LpzL . u x+2k - &c, 



In this equation put x — 0, 



.-. o = n -nu h + n ^-. u 2h -8cc, 



which is the theorem to be proved. 



With respect to the use of this theorem in expanding cos n 6 



