252 Prof. C. Niven on the Theory of an 



Xow, according to the test given above, the right-hand 

 member of this equation ought to contain differential coeffi- 

 cients of 6 only ; but inasmuch as u, v, w appear only as con- 



, . , . t dnr 1 dm* d-GTo . . . . 



tamed m -sti, oto, otq, and as -, 1 — = 1 — =— =U, it is obvious 



7 " ; ; dx ay dz 



that the right-hand member contains differential coefficients of 

 -— ^ + -j-2- + —~ only. Writing, therefore, for a, b, . . ./their 

 values, we have 



A comparison of the coefficients of like terms on both sides 

 of this equation readily furnishes the following relations among 

 the coefficients of J, 



= K 15 =K 16 =K 24 =K 26 =K 34 =K 35 ; 



K 56 = K 65 = — K 14 , K 46 = K 64 = — K 25 , K 45 ==K 54 = — R 36 ; 



2K 1 =2K n = 2K 12 + K 66 = 2K 13 + K 55 , 



2K 2 = 2K 22 = 2K 21 + K 66 =2K 23 + K 41 , 



2K,= 2K„= 2K 31 + K 55 = 2K 32 + K u , 



lv 4 = iv 42 = K 43 = K 41 — ivi 4 , 



K 5 = K 51 = K 53 =K 52 -K 25 , 



xi 6 = K 61 = J\ 62 = _K 63 — iv 36 . 



Substituting these relations in J, we obtain 



J = (K X A + K 2 B + K 3 C + K 4 D + K 5 E + K 6 F)(a + b + c) \ 



-±K u (Bc + Cb-2Vd)-iK 55 (Ca + Ac-2Ee) 



-iK 66 (A5 + Ba-2F/) f(12) 



-K M (E/+F*-Ad-Da)-K 25 (Fd + D/-B«-E6) | 



-K 36 (D« + Ed-C/-Fc). J 



But this equation may be greatly simplified; for, in the first 

 place, a + b -f c = 0, and, moreover, Be 4- Cb — 2Dc/, . . . , 

 De + Ed-C/-Fc are cogredient with X 2 , Y 2 , . . . XY, where 

 X=yf— zvjf Y=z£—xZ, Z = xrj—y^, and where A...F are 

 cogredient with x 2 . . . xy, and a . . ./ with f 2 . . . £77. But X, 

 Y, Z are themselves cogredient with lines, being the compo- 



