TRANSACTIONS OF SECTION A. 561 



For the radii of the circular arcs we have (calling a radius bar unity) 

 0A=1, 00-2 cos i (a + i3), 



OD = v^ |ot« + 1 + 2 »i cos (a + ^) I , 



OB=-v/|l+ i +? cos(a + ^) ). 



OA is a side of a rbomhus, and 00 one of its diagonals, OD is the length of the 

 largest kites, and OB the length of the inner kites. 



Any two of these four radii may be equal, except that OD is always greater 

 than either 00 or OB. 



When OA = OB we have 2 cos (a + /3)= — l/wi= —sin ^ sin a; whence 

 2 a + jS = 180°, cos a = sin ^ /3 = l/'(2 ?»). 



OA = 00 gives a + ^ = 120°. 



A = OD „ cos (a + iS) = - m/2. 



OB = 00 „ cos(a + ^)=-(m+l)/'(2»i). 



Of the two twenty-bar frames exhibited, one has joints at both B and D, with 

 1)1 = 2 ; the other has no joints at B, and its ends can be made to overlap so much 

 as to give a three-rayed star. 



4. On the Addition Theorem. By Professor Mittag-Leffler. 



Professor Mittag-Leffler called attention to the intimate relation which exists 

 between the modern theories of ordinary non-linear differential equations and the 

 addition theorem. He explained how the theories created by Fuchs, Poincart?, and 

 Picard may be generalised by making use of the considerations introduced by 

 Weierstras.<<, and showed the direction which this generalisation must take. He 

 also pointed out that the addition theorem itself may be generalised to a very con- 

 siderable extent, and that the resulting theory has important applications to the 

 theory of differential equations. 



5. Ifote on a General Theorem in Dynamics. By Sir Robert Ball, F.R.S. 



The following general theorem establishes a relation which characterises the 

 particular type of screw-chain homography which is of importance in dynamics. 



Let a, ^, y, &c., be a series of screw-chains about which a mechanical system 

 of any kind with any degree of freedom can twist. 



Let T] be the impulsive screw-chain which, if the system were at rest, would 

 make the system commence to move by twisting about a. 



Let ^ be the corresponding screw-chain related to /3, and f to y, &c. 



Then the two systems of screw-chains, a, j3, y, &c., and r), ^, C, &c., are homo- 

 graphic. 



But this homography is not of the most general type. It was only lately that 

 I succeeded in ascertaining the further general condition that the screw-chains 

 must satisfy. 



Let ■=r„f denote the virtual coefficient of the screw-chains a and ^ ; i.e., let this 

 symbol denote the rate at which work is done by the unit of twist velocity about 

 a against the unit wrench on ^. 



Then every three screws n, 0, y in one group are connected with their three 

 correspondents r], ^, ( in the other group by the relation 



1894. 



