1871.] Theory of Reciprocal Surfaces. 25 



and that in consequence the coefficients of x are erroneous in several others 

 of the formulae. Correcting these, and at the same time introducing the 

 terms in w, and writing down also the terms in as they stand, we have 



4i~ ... — 2 X + 30 —3(0, 

 24/= . ..— 6 X + 90 — 9o>, 

 2<r= ... — d — co, 



8 P = ... +6 X -90 + 9w, 

 Sk= . ..-6 x + 170— 25w, 

 2^= ... +6 X -99 +I5w, 

 8»'= ...-3O x + 210-45w, 

 c'= . . . -12 x +lO0-2Ow. 



The equations of No. 1 1 used afterwards, No. 53, should thus he 



4i+6r=s (5» — 12> — 18/3 — 5y— 2 x + 30 — 3« 



— 24^-8j+18r=(-8»+16)5 + (i5«-36)c-34j3 + 9y + 4./-6x + 90-9w; 

 and from these I deduce 



44y+fr=(44»-88)5 + (f^^3)c-f^-^ y -132^ 

 ~87/-22;- 2 - 1 x +f0. 



6. In No. 32 we have (without alteration) 0=16 ; but in the application 

 (Nos. 40 and 41) to the surface FP 2 + GR 2 Q 3 = we have 0=0, and there 

 are w=fpq off-points, F = 0, P = 0, Q=0, and X—9V1 close-points, Gr=0, 

 P=0, Q=0, The new equations involving w are thus satisfied, 



7. I have ascertained that the value of /3' obtained, Nos. 51 to 64 of 

 the memoir, is inconsistent with that obtained in the "Addition" by consi- 

 deration of the deficiency, and that it is in fact incorrect. The reason is 

 that, although, as stated No. 53, the values of two of the coefficients D, E 

 may be assumed at pleasure, they cannot, in conjunction with a given sys- 

 tem of values of A, B, C, be thus assumed at pleasure ; viz. A, B, C 

 being =110, 272, 44 respectively, the values of D, E are really determi- 

 nate, I have no direct investigation, but by working back from the 

 formula in the Addition I find that we must have D=~, E=315 ; the 

 values of the remaining coefficients then are 



I?=4 3 , G=-f, H= 

 or the formula is 



/3'=2»(»— 2)(1 In— 24) 

 —(Ufoi— 272)5+44$ 



1005 

 4 » 



I= — 198 ; 



_(f*-315)c + 



63 



+ 715/3+ 



— AC — gT$ — set — \j — /i X — r0 — fio 



— tiC —g'B' — x'i' — Xf — /i' x ' —r'ti'—f'bi'i 



