OF SIMULTANEOUS ORDINARY DIFFERENTIAL EQUATIONS. 537 
Aa + By? + Cy,” = 1, 
ce yee eye = 1. 
The first of the differential equations may be written 
(A + Bp,? + Cp,*) (Az? + By,? + Cy? — 1) = (Aw + By,p, + Cynp,)’. 
One of the integrals is, therefore, 
(A + Be? + Ce,*) (Av? + By,? + Cy? — 1) = (Aw + Bye, + Cy.¢2)°. 
Thus, one of the equations that give the singular solutions is 
(Be, de, + Ce, de.) (Ax? + By,?+ Cy,? — 1) = (By, de, + Cy, dey) (Ax+ Bye, + Cyc). 
Unless 
Aa? + By? + Cy”,—1=0 and Ax + By,c, + Cy,c, = 0, 
this may be reduced to 
(Be, de, + Ce, de,) (Ax + By,e, + Cyy¢.) = (By, de, -+ Cyz de) (A + Be,? + Ce,?). 
This last equation only contains @, y;, y2, in the combinations y, — cx, yy — Cx, 
and the same will be trve for the other equation of the same form that may be 
derived from the second -quation of the complete primitive. 
From these two equations and those of the complete primitive we can eliminate 
the ratio dc, : dc,, and the expressions y, — ¢,” and y, — ¢,x#, so as to have an equation 
in ¢, and ¢, only. In accordance with § 14 the first singular solution thus given will 
be included in the complete primitive, and the only proper first singular solutions are 
given by taking 
ax? + by, + cy? —1=0 and ax + bye, + cyoc, = 0, 
or 
Aa? + By,? + Cy, —1=0 and Aw + Bye, + Cy,c, = 0. 
§ 23. In order to integrate for the first singular solution, it will be useful to 
transform the equations to others in terms of ¢, uv, v, the values of «4 which satisfy the 
equation 
Aax? Bby,? Coy,” 1 
a+ pA Dee (ds SoA eae 
MDCCCXCV.—A. SZ 

