TRANSACTIONS OF SECTION A. 463 



5. On the Modular Curves. By Professor H. J. S. Smith. 



6. On tlie Principal Screivs of Inertia of a free or constrained Bigid Body. 



By Professor R. S. Ball. 



7. On the Applicability of Lagrange's Eguations to certain Problems of 

 Fluid Motion. By Professor J. Purser. 



8. On the Occurrence of Equal Boots in Lagrange's Determinental Equation 

 of Small Oscillations. By Frederick Purser, M.A. 



This paper is an endeavour to supply a more elementary proof of the important 

 result which M. Soraoff claims to have been the first to establish* — that the 

 existence of equal roots in Lagrange's determinantal equation of small oscillations 

 does not affect the stability of the system. It is further shown that the same con- 

 clusion holds for a certain system of differential equations of the first degree having 

 also a physical application. 



The solution of the equation of small oscillations — 



-^± = a n x 1 + a li x. i a lu .v n , 



_ 2 = rt 2r r 1 +.fl 2 „.r 3 a 2ll x n , 



&c, &c. 

 (where a 1 o = n 21 , &c.) 

 as obtained by the method of indeterminate multipliers is — 



X a = *ili ffi + a?sl 2 m + &c. = A 1 sin (>/£» t + e 1 ), 

 X 3 = *!&.« + x 2 £ 2 <» + &c. = A 2 sin ( </&® t + e"), 

 &c, &c. 



Where £, m , | a <», &c, £» £®, Sec, are the systems of values of £„'£„ &c, which 

 satisfy the equations, 



(«11 + *0£l + «12^2 + &c - = °> 

 ftjilj + (« 22 + ft)£ 2 + &C. = 0, 



&c, &c. 



Corresponding to the roots k a) , k i2 \ of the determinant in k. 



Now, by suitable transformation of this latter system it is shown that when m 

 roots of the determinental equation are equal, the common value being k, the cor- 

 responding system of £s involves m — 1 arbitrary parameters. Hence in this case, 

 on the one hand, m distinct integrals of the oscillation system are merged in one 

 X = A sin ( VT t + e) ; on the other, thisone, in virtue of the m - 1 arbitrary para- 

 meters implicitly involved in its left-hand member, is equivalent to m distinct 

 integrals. 



X. 1 = A. l sm{s/kt + t 1 ) X„ = A 2 (V£t + e.,)> 

 &c. 



The occurrence of equal roots in the determinantal equation therefore only 

 diminishes the number of distinct vibration periods, without affecting either the 

 periodic form of solution or the number of distinct integrals. 



A second method is employed by the author, based on the theory of orthogonal 



* " Sur l'qeuation algSbrique a l'aide de laquelle on determine les oscillations tres- 

 petites d'un systeme de points materiels." — ' Memoires de l'Academie de St. Peters- 

 bourg,' vii. serie, tome i. No. 14. 



