THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 245 



we put 



4P (v) r dt* 



h y + 1 ' v/TVfjj = 



and now the expression for the catenary can be reduced to 



N (tan 0c*')" + * = (B + E,t + B/- + . . . + B^ 2 ''-') ,/T, 



+ { (B - B!< + B 2 2 - ... - B 2( ,_/-i) v/T 8 (8). 



Thus for 



/*=?, o: = z(l-z) 3 , y = z(l-z) (9), 



P (r) = 5 * 4 = for z = - -I + 1 v/21 (10) ; 



and we can calculate M, A, h, and the six B's in 



N (tan|0e*')= = (B + B L + . . . + B/) V /T, 



+ t -(B-B i < + .. --B/) X /T. : (11); 



this catenary has been drawn stereoscopically by the late Mr. T. I. DEWAR. 



21. ABEL'S integrals are applicable immediately to the construction in polar 

 co-ordinates r and 9 of algebraical orbits or catenaries under a central attraction of 

 the form (Huco GYLDEN, ' Kongl. Svenska Vetenskaps-Akademiens Handlingar," 

 1879) 



(1); 



with p. = the orbit can be realised by two balls, connected by an elastic thread, 

 whirling round each other in the air ; and the addition of a term to P, varying 



inversely as r 3 , merely has the effect of qualifying the angle 9 by a factor. 



i 

 Putting r = -, and denoting the velocity in the orbit by v and twice the rate of 



\Ju 



description of area by h, 



(2), 

 4 + 2 Aw. 3 + Bit 2 - 2Cw - D = U (3), 



and now put ;t = z ^ to identify with ABEL'S results. 



