606 KEPOET— 1887. 



be two linear diiFerential equations of the nth order, aud let .r and t be connected 

 by the equation 



dx dt 



P = Q (17) 



where P and Q are functions of x and t respectively, Sir James Cockle has 

 assumed 



li=(Q"(0}'s^ = {V'«(0}'' .... (18) 



Let V be a third independent variable, and assume 



(l,V,.V„...V„)(£,l)V = .... (19) 



(l,W„W„...AV„)(|,iyV = . 



(20) 



where (19) is connected ^vith (15), and (20) with (16) by the relations 

 dx ^ dt ^ 

 dv-^>do = ^ (21) 



Equations (19) and (20) become identical when 



V. = W„V, = W„. . . V„ = W (22) 



a system sufficient to connect (15) with (16) by (17). 



By this method the criticoids have been calculated for n = 2, S, 4 respectively, 

 aud the following results obtained : — 

 n = 2. 



cl>^;\x) + 4<^,(.r)(/..,(.r) _ x/^,<'Yf) +4x/.,(0V.,(0 



{^MV^ ~ {^tm • • • ^^^> 



n = 3 



and, 

 n =4 



{<^s(a=)i^ "~ {Mm ■ • • ^"^^ 



(28) 



i(f>\'Hx) + 8(A,(x)(/,/x) 3,j./"(0 + 8yj,,( t)yl,,(t) 

 22(|,.(.r)^ + 1 20,"'(: f) -27.^,(x) 22xj..(0^ + 12>/.,'"(0 -27x/.,(n 



{<i>M?- ~ {f.am • ('^'* 



.c»)(^.) + 2<^.(A)(^,»'(.r) 1 6<t>,( x)4>,(x) - f ,j>,(.rV -6<^3(.r) 



_ yl^rCt) + 2^/.,(0^f^/»(0 + 6^/., (0^/.,(0 - ^,°r|..(0=' - 6^.3(0 



The results (23) to (28) are included in the general formulae given by Mr. Harlcy 

 in the paper above cited. 



5. Complete Integral of the n-ic Differential Resolvent. 

 By the Rev. Robert Haelby, M.A., F.B.S. 



Representing the roots of the n-ic algebraical equation whose coefficients are 

 functions of a single parameter (c) by j/j, y^, . . ■ y„, the complete solution of its 

 differential resolvent is 



^1^1 + ^22/2+ • • • +^nyn, 

 where c^,c„. . . . r„ are independent arbitraries subject only to the condition 

 Cj + Cj-H . . . +c„=l. 



