540 EEPOET — 1886. 



tion is a particular case of the correspondence, and the formulae (2) are probably 

 the simplest obtainable for projection. 



5. On the Sum of the nth Powers of the Terms of an Arithmetical Pro- 

 gression. By Professor R. W. Genese, M.A. 



We know that 



fn+l 



1" + 2" + 3" + . . . + f" = — + lower powers of t 



n + 1 



= 0(0 say. 



Then(_t + iy + it + 2)"+ . . . (t + ry = (f>(t + r)-(})(t). 



Now this statement is true for more than n values of t (viz., for all integral 

 values). 



Therefore it is an identity. 



Putting ^ = _- we obtain 

 a 



(a + (^« + (rt + 2<i)" + . . . +{a + rd)n = d'' \^(^+r^-^ {~|)1 

 A number of interesting results can be obtained from the identity. 



6. On a Form of Quartic Surface with twelue Nodes. 

 By Professor Catlet, LL.D., F.B.S. 



Using throughout capital letters to denote homogeneous quadric functions of the 

 co-ordinates (x, y, z, w), we have as a form of quartic surface with eight nodes 

 S2 = (* J{ U,V, W)^ = ; viz., the nodes are here the octad of points, or eight points 

 of intersection of the quadric surfaces U = 0, V = 0, W— ; the equation can be by 

 a linear transformation on the functions U,V,W (that is, by substituting for the 

 original functions U,V,W linear functions of these variables) reduced to the form 



Suppose now that the function Q can be in a second manner expressed in the 

 like form a =P- + Q2 + R^ (where P, Q, R are not linear functions of U,V,W) ; 

 that is, suppose that we have identically U- + V^ + W* = P^ + Q'^ + R^, this gives 

 [j2_p2 + V^-Q2+W*-R- = 0; or, writing U + P, V + Q, W + R = A, B, C, and 

 U - P, V - Q, W - R = F, G, H, the identity becomes AF + BG + CH = ; and this 

 identity being satisfied, the equation Q = of the quartic surface may be written in 

 the two forms 



Q = (A + Fr- + (B + G)« + (C + H)^ = 0, and 



G = (A-F)* + (B-G)* + (G-H)2 = 0; 



viz. the quartic surface has the nodes which are the intersections of the three 

 quadric surfaces A + F = 0, B + G = 0, C + H = 0, and also the nodes which are the 

 intersections of the three quadi-ic surfaces A — F = 0, B — G = 0, — H = 0. We may 

 of course also write the equation of the surface in the form 



fi = A2 + B"- + C^ + F^ + G- + H^ = 0. 



An easy way of satisfying the identity AF + BG + CH = is to assume 

 A, B, C, F, G, H = ayz, bzx, cxy,fvw, yyw, hziv, where the constants a, b, c,f, g, h 

 satisfy the condition af+ hg + ch = ; this being so, the functions A, B, 0, F, G, H, 

 and consequently the functions A + F, B + G, C + H and A — F, B-G, C — H each 

 of them vanish for the four points {y = Q,z = Q, ?<; = 0), (z = 0, .r = 0, w = 0), {x =0, 

 y = 0,w = q), (.r = 0,y = 0,3 = 0), or say the points (1,0,0,0), (0,1,0,0), (0,0,1,0), 

 (0,0,0,1) ; it hence appears that the quartic surface 



Q. = a^y'^z^ + IPz'x^ + c-x'^y'^ +f^x-io'^ + g'^y'^w'^ + h'^z'^w'^ = 

 is a quartic surface with twelve nodes ; viz. it has as nodes the last-mentioned four 



