508 ON THE RECIPROCATION OF A QUARTIC DEVELOPABLE. [372 
and suppose for a moment that the determinant formed therewith is = K; suppose also 
that the reciprocal matrix is 
81, ©, 8 . 
& 8, SR 
s, e, 91 
s, m 91, 3) 
Then we have 
if8a + dA (21« + Sfry + ®z + 2w) + A (2lc?« + <£)<% + ®dz + idle) = 0, 
with the similar equations involving b, c, d and (.£), 23, 5, 931), (©, (5, 91), (2, 931, 91, 3)) 
respectively. 
But, substituting for («, y, z, w) their values from the equations (2), it is easy 
to see that we have 
21« 4- toy + ®z + 2w = — Ka, 
the last-mentioned equation thus is 
K (^8a — ^ 8A^ + A (218« + «£>83/ + ®8z + 28w) = 0. 
But K is = 27 □ (see my paper “ On Certain Developable Surfaces,” Quarterly 
Mathematical Journal, t. vi. 1864, pp. 108—126, [344]), which is =0, and we thus have 
218« + tQBy + ®Sz+ 28w = 0, 
and similarly 
¿58« + 23 By + %Bz + 9318w = 0, 
©8« + %Sy + $8z + 918w = 0, 
28« + 91182/ + 918^ + 3)8w = 0. 
But observing that in virtue of the equation K = 0, we have 
21 : £ : © : 2 = £ : 23 : $ : 931 = ® : % : g : 91 = 2 : 9J1 : 91 : 3), 
these are, as they should be, one and the same equation. 
The values of the coefficients 21, 23, &c. are given, p. 112 of the paper just 
referred to, viz. writing □ in place of TJ, we have 2l = 3X 2 —4a 2 D, &c. where 
X = a?d — 'Babe + 28 3 , 
Y = abd — 2 ac 2 + b 2 c, 
Z = — acd + 2¥d — be?, 
W = — ad 2 + 3bed — 2c 2 ,