MATHEMATICAL SECTION. 129 
(13) will take sia of the following annuals forms : 
0=42%,—- 4) = {+ Ac (2,— f 2) Fe + Ay (@, — ~ 720) (13"} 
0 = ay (2, — a2) + az(a,— fine tar, — ree (13%) 
The following relations will be much referred to: 
Y2 ey ah Yx 1 
(teat ete (16) 
Ea il Sie ie I iiaale SG eee 7 
x, Yx e, ae %, ; UP z, x, ce x, ‘ 
Z,.—2 r 4 
x y toe yy, eae yx (17) 
2 Bm ae a 
0=42, ¥, Ze Ys 2, &, (18) 
Replacing in these Ax, Ay, Az; Y,, 25) 3 Zar Uys Ye 
Des A ii yak 
by Gr Ger i wr Borie 5 Ses Bar Fo (19) 
2 fy 2 2 2 2 2 
Nn 2 “a i B a, 
we have: 0 = ae Be =a oh “ip aa (16' 
By Sar i — is — 8, a Tic im a," a a,” wd Pa 
By ieee A OR, “leat oti 
a,” ae 8, ee B, os dy” (eu) 
a, ie nes Vv 
0 Saw 7 tn a,” = ay” a? ie (18") 
and these relations also will be found correct. 
Because in the equation of the diorthodic surface the terms in 
x,y, 2 are wanting, there must be lines, perpendicular to the co- 
ordinate planes, lying wholly in the surface. To determine those 
perpendicular to the zy- plane, I place = 0 the term in ee de- 
pendent on z and that in (13") independent of 2, or 
Ay 
> A 
Be 2 
O= — yar art n— 7. &) 
to % vy 
= — part ;— 2) by (16) and (16! 
4 
O = ayz, - a+ az(2, — 20) Fs 
: iG as 2) y by (16) and (16') 
=> 
