32 Mr H. F. Baker, On the Concomitants [Nov. 25, 



the results obtained, will show the importance of the critical 

 spheroid: — 



In any 5-constant or isotropic rotating spheroid one of the 

 principal stresses is everywhere perpendicular to the " meridian " 

 plane, and of the two in the meridian plane the greater makes an 

 obtuse or an acute angle with the perpendicular on the " polar " 

 axis produced outwards according as the spheroid is more or less 

 oblate than the critical spheroid. In particular the surface is under 

 a tangential tension in the meridian plane in the former case, but 

 under a compression in the latter. In the critical spheroid one of 

 the principal stresses is everywhere zero, and on the surface there 

 is no stress at all in the meridian plane. In any species of bi- 

 constant isotropic material, for a given value of the equatorial 

 diameter, the critical spheroid is the form in which the " tendency 

 to rupture " on Saint- Venant's theory is the greatest. 



In the case of uniconstant isotropy the character of the strain 

 throughout rotating spheroids of all shapes is completely investi- 

 gated, and is shewn in a table. In the general case of 5-constant 

 material a similar, though not so exhaustive, analysis is given. 

 Tables supply the values of the changes in the lengths of the 

 equatorial and polar diameters, and the strains at the centres for 

 various kinds of biconstant isotropic materials in spheroids of 

 various forms. 



The variations of some of the more important quantities are 

 also shewn graphically. 



(4) On the concomitants of three ternary quadrics. By H. F. 

 Baker, B.A., St John's College. 



[Abstract] 

 The author applies a modification of the symbolical method 

 suggested by Clebsch and Gordan (Math. Annul. I. 90 and I. 359) 

 to obtain the set of concomitants in terms of which all the system 

 are expressible as rational integral algebraic functions. The 

 result is given by the following table. The forms are taken 

 respectively to be 



a 2 - a ' 2 = a " 2 = 



U x U x U x '"> 



r 2 — c ' 2 = c " 2 = 



V x — V x — L x — 



Also (aa'uf is abbreviated into iC = u/ = u a - z =...', 

 and (aa'a") 2 into a a etc.: 



and so for the other two forms. 



Such a symbol as (523) preceding a form indicates that the 

 form is of the fifth degree, second class, and third order. 



