146 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Hence 
Also 
dt ^rOr-pdp ds _P d 1 
■\/(r z -p z ) r 
ds j - ^ds 2 
dt x = 
dt n 
= dscy - -ds.. 
etc., etc. 
ds k — dt k 
dsi — dt t 
ds k+ 1 
ds l+ 1 
( 7 ) 
. ( 8 ) 
Cor. Owing to the homogeneity, of degree zero in p and r, of the 
expressions for ds, ds 1 , . . . ds m ; dt, dt x , . . ., it follows that any linear 
homogeneous equation in these is immediately integrable in terms of 
p and r. 
§ 61. Maclaurins Theorem. 
Let us seek to determine the curves for which 
ds* = Ads + B dt 
A and B being constants. 
Here 
Hence 
In (2) put 
2 ¥-ds 1 - ^- 2 ds = Ads + B dt. 
2— dp - ~<dr = A dr + B (dr - Tdp\ 
r r 2 \ r J 
p = ry, 
dr 
= dy{2 + B)y/{A + B - (1 + B)y 2 } . 
the integral of which is 
In particular, when B = — A 
r{A + B-^(l + B)}|^ = C 
or 
r(l-Ay 2 f- 2A = C', 
The corresponding equation to s, is then 
( 1 ) 
• ( 2 ) 
• (3) 
• W 
• (5) 
