18 REPORT 1875. 



that for values of c between 30° and 150° the curve is simplex, and there is no 

 nodal point. 



In particular, if the cyclic arc and its satellite are at right angles to each other, 

 the cubic is simplex and trilateral or campaniform. 



At the terminal values, c=+30°, the cubic is cuspidal, and will be separately 

 considered. 



If 4 cos ^O'S, the cubic may have all four forms, according to the value of the 

 parameter k. 



The transition from the simplex to complex genera takes place at the critic centres. 



The discriminant (64 8^— T-) varies as 



{ 3/c- ?-^ (9-8 cos «c) } '- I (i cos=c-3). 



The cubic is complex, nodal, or simplex, as the discriminant :> = <;0. 



5 

 £x. cos c= ■ ,„ : the conditions become 

 4V2 



("-472)0- 972)^ = <^- 



27 "^8 



Between the limits of .--^ ,^ and ~of-jn foi" " the discriminant is positive and 



the curve is complex ; at the former limit acnodal, at the latter cnmodal, beyond 

 these limits simplex. 



In particular, ii k = ^ , the invariant S=0, and the cm've is simplex neutral. 

 o 



7. Cuspidal forms, — At a cusp both invariants S', T are equal to zero. Hence 



1 = 2kcosc : 3k^— 6k cosc+2=0. 

 In this case 



K = + -J-g, c = 30° or 150°. 

 The corresponding trilinear equations exhibit the cusps 



(V3 + '^)' + ¥ =^' {c=m°), (1) 



(-73 + ^)' + ? =^' ^'=^^^°^ (^^ 



In (1) the cusp is 30° distant from A and 60° from B5 in (2) it is 30° from A 

 and 120° from B, and intermediate. 



8. By dualizing, corresponding theorems may be obtained for cubics of the third 

 class with triple foci and triple cyclic arcs. The properties of plane cubics of this 

 class may be deduced, although, as in cubics of the third order, they are not co- 

 extensive. 



There are three species only in piano of trifocal cubics (complex, bitangential, 

 and simplex), whereas all five occur in trifocal spherical cubics. Plane trifocal 

 cubics exhibit (in their point-equations) cusps at infinity, but have not coincident 

 asymptotes. 



9. The other two groups of spherical cubics are reserved for future consideration. 



Elementary Solution of Huyghens's Problem on the Impact of Elastic Balls. 

 By Paxil Mansioit, Professor in the University of Ghent. 



1, If two positive quantities x, = r—z, and y have a constant product )-^, their sum, 



r~z +-!—=2r+^^, 

 r — a r — z 



is the smallest possible when 3 = 0, viz. when cc=y=r. 



