86 



A very general solution of this etiiiation comes from a::^h=-=c, j'= ij = fi 

 = 0, which are tlie difierential equations of the series of spheres that pass through 

 a given fixed circle, including, as particular cases, concentric spheres, planes inter- 

 secting in a fixed line, and parallel planes. 



It may be shown that the a-bove equation factors into four factors of the form 

 l'6"— c' D -~ \ c^—a^ J/' = 1 a'— 6' N^ where a\ b\ c\ are the roots of t!ie cubic 

 found by replacing «, b, c in the above matrix by a — x, b — x, c — x. The diflferential 

 equation may also, by the usual reciprocal transformation X=^L, Y=^ M, Z=^ y, 

 U -\- u = L X -{- M y -{- N z, be reduced to a simpler form. 



The preceding diflferential equation and the resulting theory of orthogonal 

 surfaces were obtained by quaternion analysis. Briefly, if '/, /6, are two perpen- 

 diculars to the surface normal 6, that are also surface normals, then we have, 

 (1) 5^<T — o; (2) ;SZyA = o; (3) ^'A(7yA(7=zo. 

 We may replace (3) by 



(3') S?.ax'i^(^i = o,or S /. (}> V(y?. = o, vfheve <i/. = l (ri'S?- (^1+ ffi'S?. Vi). 

 Thus <p is the self conjugate linear vector function, whose matrix is given above. 

 From (1) and (3\) we find 



F?. Voo Vcy=o 

 This determines /I as one (and A a as the other i of tlie two latent directions of the 

 plane self-conjugate vector function V (^ <^ V(t/-. There is therefore in general but 

 one pair of normals that may satisfy the conditions of which (2) becomes a condition 

 upon (T, or the differential equation satisfied by /(/, y, z) in order that it may possess 

 a pair of orthogonal conjugates. If, however, the above plane vector function have 

 equal latent roots, then its latent directions become indeterminate. This means 

 that (1) becomes a factor of (3M so that the only equations to be satisfied are (1), 

 (2). These may be satisfied without other condition upon o than the above equality 

 of latent roots which is tiie difterential equation that we have given at the be- 

 ginning of the paper. 



XoTE. — Since presenting the above I have noticed that the latent roots of the plane strain 

 mentioned are proportionals to the principal radii of curvature of normal sections of the 

 surface f (x, y, z] =c. The above differential equation o£ second order therefore expresses 

 that every point of each of these surfaces is an nmbilic. Hence the general solution consists 

 of a system, of spheres (or planes) with one variable parameter. tu = Ax, y, z). The above 

 qtiatemion method gives also the conditions that a system of linos may be the intersection 

 of one pair of orthogonal systems of surfaces, or of an infinite numlier of such pairs. 



The Calendar Grox'p. By C. A. Waldo. 



