534 
Proceedings of the Royal Society 
The equation (16) now takes the form 
(^-M 3p2 -(^ + ^) 2 + M 1 ( = 0) 
- 2 (m 2 + u)$p<j>i'<l>f 
(21) 
where (m 2 + m) depends on ( — - + p 2 \ by 
m. 
u 
( 22 ) (m 2 + ur = M 2 -p* + 2(f- + pf 
4 . 
The relation (21) depends through (22) explicitly on (^~ + p 2 J 
and on p only. The term (<f>'p) 2 = S pcfxfi'p = S p-ap depends on p only 
and on the constant vectors <£/, because we have 
<f>p = — %i'Si<p'p = — ^i'Spcjii' ; 
hence 
(23) <£<£'/) = *rp = — (<jii'$p4ti' + cfyj'Spcfyj'). 
7Yl> 
Through (13) and (14) the quantity — - contains the three arbi- 
'll/ 
traries of the question, namely, u, and the two independent para- 
meters of the operator p( )q. 
The equation (21) represents now, by the extremity of p, any 
point on the resulting force of the system in any of its positions. 
If we submit the three just named arbitraries to a relation of 
condition, then we “ pick out” so to say a particular point on every 
one of the resultants, and the aggregate of these points will form 
a definite surface. 
The eqitation of condition which gives the surface of the lowest 
771 
possible degree will be of course — - + p 2 = a constant parameter. 
u 
For convenience sake we put 
(24) 
h + Oht + p z = o 
U 
g + m 2 + u = 0 , 
— h being the parameter introduced ( h being identical with the x 
