85 



From this expression the foUowing table has been computed for a 3-inch 

 shaft running at 100 and 250 revolutions per minute: 



It is worthy of remark that in long lines of shafting the influence of belt 

 pull on the bearings is very slight compared to the weight of shaft and pulleys, 

 so that the loss in friction is but little more than that due to weight alone. 



AVith better alignment and better lubrication the loss will be less than that 

 here given; in long continuous lines of shafting the bearings are always more 

 or less out of line, and for this reason the loss will be less if short lengths be 

 employed. 



Orthogi:)Xal .Sukkaces. By A. S. Hathaway. 



It is well known that a given system of surfaces/" ix, y, z) =^ c h.ns in general no 

 pair of orthogonal conjugate systems, /. e., sucli that the surfaces of the three sys- 

 tems through any point are mutually orthogonal at that point. It has been shown 

 by Cayley [Salmon's Three Dimensions, p. 417] that /(x, y, z) must satisfy a dif- 

 ferential equation of third order if it possess a pair of orthogonal conjugates. In 

 the course of some recent investigations on fluid motion I was led to observe that a 

 given system of surfaces migiit have two pairs of orthogonal conjugates, in which 

 case it would have an infinite number of such pairs. In order that such may be 

 the case/(x, v, 2^ must satisfy a differential equation of second order which is a 

 particular integral of Cayley's equation of third order. This difierential equation 

 is, in Cayley's notation, 



[( a, 6, c. /. fi. h 1 ( L, M, xV l'^ — 1 a -f 6 — c n J.^ - .1/' - .V - )] - 

 = 4(1-- M- -L y-) {A, B, C, F, G. H.\ i L, M. X V 

 where L, M, jV, a, b, c, /, 7, h, are the first and second diflferential coefficients of 

 fix, y, z), and A, B, C, etc., are the minors of a, b, c, etc., in the matrix 



