182 Mr. J. Cockle on the Solution of certain Problems. 



the British Association for the Advancement of Science, to 

 which, for the purpose of avoiding prolixity, the author begs 

 leave to refer the reader of this paper. It will be seen that, 

 using the notation there employed, 



ro(0, 1, 1) = 1 +m{\, 1) = 4, . . . (336'.) 

 ?»(0, 0, 1, 1) = 1 + w (1, 1, 1) = 6, . . . (337'.) 



since m(l, 1, 1) = 5, (307.) 



m (0, 0, 0, 1, 1) = 1 + ffi (1, t, I, 1) = 12, . (338'.) 



since 02(1,1,1,1) = hi j (317.) 



10 may however be written for 11 in this last equation, for 

 reasons which will be manifest on referring to page 34-8 of the 

 above-mentioned Report ; so that we may make 



0z(O, 0,0, 1, 1) = 11 (339'.) 



So m(0, 0,0, 0,1, 1) = 1 +m (1,1, 1, 1,1) = 48, . (340'.) 



for w(l,l, 1, 1, 1) = 47; -(327.) 



and 



m (0, 0, 0, 0, 0, 1, 1) = 1 + m (1, 1, 1, 1, 1, 1) = 924, (341'.) 



since m (1, 1, 1, 1, 1, 1) = 923 (328.) 



Now each of the newly-valued functions may for convenience 

 be represented by m (0 r ~ 1 , l 2 ), — a very obvious abbreviation. 

 Let n(0'' -1 , l 2 ) denote the corresponding function when, in- 

 stead of the process to which the above equations refer, we 

 apply that which the writer of this paper has used in the last 

 two volumes of the present series of this Magazine. Then, 

 by investigations similar to those pursued in the particular 

 cases already therein treated, he has been conducted to the 

 general condition, 



n(0 r ~\ l 2 ) = 2r (".) 



This last condition, which the author hopes to discuss on 

 some future occasion, gives 



9i(0, 1,1)=4, (336".) 



«(0,0, 1, 1) = 6, (337".) 



n (0, 0, 0, 1, 1) = 8, . . (338".) or (339".) 



w(0,0, 0, 0, 1, 1) = 10, (340".) 



0(0,0, 0,0,0, 1, 1) = 12 (341".) 



On the condition marked (338".) or (339".) is founded a pro- 

 position announced at p. 405 of the last (44th) volume of the 

 Mechanics' Magazine. The more general one, which will be 

 found at p. 36 of No. 1196 of that work, is based upon the 

 above condition (".) 



Of course the consideration of the above propositions will 

 require some extensions of the formulas given by the writer 



