1 
[719 
720] 55 
otation of A 
eless that, to 
11 (the throw 
7r, where the 
by a wheel 
station of A' 
ind therefore, 
ice, \ being 
Is in E an 
viz. E moves 
' successively 
(ular velocity 
therefore n) 
720. 
NOTE ON ARBOGAST’S METHOD OF DERIVATIONS. 
tween guides, 
4ii a toothed 
Then if a. 
imp upon L, 
mt; whereas, 
ct will be to 
equently the 
and E the 
lish between 
[From the Messenger of Mathematics, vol. vn. (1878), p. 158.] 
It is an injustice to Arbogast to speak of his first method, as Arbogast’s method*. 
There is really nothing in this, it is the straightforward process of expanding 
cf) ^ a + bx -f- j—— cx 2 + ... ^ 
by the differentiation of cj>u, writing a, b, c, d, ... in place of u, ^> & c - or 
say in place of a, u', u”, u", &c. respectively ; thus 
rack Z) with a 
in gear on the 
other of them 
micate to £ a 
A or rotation 
re. This is in 
>wer instead of 
<pa, fia. b, ^ {<f>'a. c + fi'a . ¥], % Ccf>'a. d + <£"a . be ) 
[ + (f>"a. 2 be + (f)"a. 6 3 J 
= ^ {(f>'a . d + cp"a . Sbc + ft 'a . b s ], &c., 
and in subsequent terms the number of additions necessary for obtaining the numerical 
coefficients increases with great rapidity. 
That which is specifically Arbogast’s method, is his second method, viz. here the 
coefficients of the successive powers of x in the expansion of </> (a + bx + cx 2 + dot? + ...), 
are obtained by the rule of the last and the last but one ; thus we have 
cf>a, cji'a.b, cji'a. c + cj)"a. ^b 2 , <£'a. d+ </>"«. 6c 4- (¡)"'a.^b z , &c., 
where each numerical coefficient is found directly, without an addition in any case. 
See Messenger of Mathematics, vol. vn. (1878), pp. 142, 143.