558 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
(hjr 0 t , 7. _IT 1 
1 dx + 3 hi + 3 & + 4 a w ~ \ 
or \ ;.(164*) 
Sf dyfr , d\fr | 
x w+y ar + z af+ H ’5:= K j 
where , as usual, means the partial differential coefficient of x(j with respect to x, 
{k x , k 2 , & 3 , & 4 ) being afterwards put for ( x , y, z, w), and where K is some constant. 
150. If S be any small circle, we may define the coordinates of S with respect to 
the system (1, 2, 3, 4) as constant multiples of the powers ; thus, denoting them by 
rj, £, oj, we will take 
k~kyi r S|1 , rj—k.yTTs v, C = k'3- 7T s,,3> w — ^4 ,77 s,4, > 
k x , k 2 , L, k 4 having the same values as in § 149. 
Since 7r Sj 0 == 1, we see that the coordinates of any small circle must satisfy the non- 
liomogeneous linear relation 
h '% +k % +k ^+ k ^L =¥ ~ .o«5) 
151. If, however, S be a great circle, we shall have, since 7r Si <,=0, 7r S)S = — 1, the 
homogeneous linear relation, 
*'tf+*'*'*■** Sr +**t£= 0 >.< I6 ®> 
and the nondiomogeneous quadric relation, 
2^(£ y, C co)=—K .(167) 
The Small Circle .—§§ 152-157. 
152. If P be any point on the circle S whose coordinates are (£, y, £, oj), we shall 
have by the equation 
n( P ’ X ’ 2 ’ °’ 4 Wo 
‘ S, 1, 2, 3, 4 ) 
since 7r P S =0, 
lt x+ ^ y+ fs~ +d ^ w=0 . (1C8) 
Thus the equation of a small circle is of the first degree. 
It follows that the equation of the first degree, say 
ax-\-lny-{-cz-\-div—Q, 
