THANSACTIONS OF THIC SKCTION'S. .'29 



and find its logarithm before \vc eau proceed to deduce u, iioui a. ^iil these 

 operations have to be repeated to find u.^ from wi ; nay, ■we must first find c^ from c, 

 and that is still more arduous. 



But when we assume p, =^7r„~-^, as argument, all is greatly simplified. 



The relation of c, Cj, <?„, c^ .... in Lagrange's scale corresponds with p, 2p, 2^p, 2'p, 



and in Legendre's scale Avith p, 3p, o'-'p, -Tp . . . . , which involve no trouble iu 



calculating. No doubt we need tables (of single entry and easily compiled) to yield 

 c, b when p is given, and p when c is given. Presuming these, we may treat .f and 

 r(c, to) as functions of p and <a ; after which the difficulties of the constant multi- 

 plier A vanish, and Legendre's scale becomes practical to us. 



Denote — log A, i. e. — log »/(l—c- sin -/3J, for the moment, by *(/>) (here the 

 common log is intended) ; then, among the numerous series which express fimctions 

 of the amplitude oj in terms of x and p, the author selects (with X for Napier's log) 



,., , . ., ^ 1— cos2.r , 11— cos 6a; , 11— coslO.f , „ 

 ^ ^ ' sm 2p o sin 6p 5 sm lOp 



where sin p is written for \{t»—e~''). By hypothesis, F(c,^)=|F(c, -in-); hence 

 when to=i3, x=\tt, and we get, writing cosec p for the reciprocal of sin p, J'^fpl 

 =M{cosec 2p+} cosec 10p + icosecl4/)+^\ cosec 22/3-I-&C.}, M being the mo- 

 dulus of the common logarithms. 



Assuming that we have a table of *fp), then given p and co we have the equation 

 log tan ^(to,— co)= log tan to — $(p) to find q>j ; log tan 5(0)^— 0)^) =log tan cu,- *(3p) 

 to find Q)^ ; log tan \i^ai.^—oi^=.\o'^ tan ai.,—^{??p) to find w,, and so on. The 

 approximation is sufficient when *(3"p) is negligible ; and this result is obtained 

 so rapidly, that iu the extreme case of p = i, x=-2>~-ai^ is correct to ten decimals. 



To bring the method to a practical trial, the author has calculated to twelve 

 decimals a skeleton table of *(p) for p=;0-5, OG, 07, 0-8, 00, and from p = l 

 to p = 14'3 at intervals of 0-1. The table is given in the paper, and also exam- 

 ples of the method. The process also by which the table was constructed, with 

 the aid of tables of cosec p and e~^, previously calculated by the author, is ex- 

 plained. 



General Theorems relating to Closed Curves. By Professor P. G. Tait. 



The closed ciu-ves contemplated are supposed to have nothing higher than double 

 points. By infinitesimal changes of position of the branches intersecting in it, a 

 triple point is decomposed into three double points, a quadruple point into six, and 



I'^T— 1 ^ 



generally an a-ple point into —^ — — double points. (1) A closed curve cuts any 



infinite unknotted Hue in an even number of points [infinite here implies merely 

 that botli ends are outside the closed curve]. (2) The same is true if the line be 

 knotted. (3) If any two closed curves cut one another, there is an even number 

 of points of intersection. (4) In going continuously along a closed curve from a, 

 point of intersection to the same point again an even number of intersections is 

 passed. (5) Hence in going round such a closed curve we may go alternately 

 above and below the branches as we meet them. (G) By (3) the same proposition 

 is true of a complex arrangement of any number of separate closed curves super- 

 posed iu any manner. (7) In passing from the interior of any one cell to that of 

 any other^in any system of superposed closed curves — the number of crossings in 

 always even or always odd, whatever path we take. (8) Hence the cells may be 

 colom-ed black and white iu such a way that from white to white there is always 

 an even number of crossings, and from white to black an odd number. Such closed 

 curves therefore divide the plane as nodal lines do a vibrating plite. 



The above are the enunciations of the propositions proved iu the paper, which, 

 with the necessary figures &c., will be found printed in exfenso in the ' Mes.^enger 

 of Mathematics,' vol. vi., January 1877. 



