1878.] Mr. I. Todhunter on Legendres Coefficients. 381 



of the amount of current makes the strata distinct and steady. Most fre- 

 quently a point of steadiness, sufficient for photographing, is produced 

 by the careful introduction of external resistance ; subsequently the 

 introduction of more resistance produces a new phase of unsteadiness, 

 and still more resistance another phase of steady and distinct stratifi- 

 cation. 



7. The greatest heat is in the vicinity of the strata. This can be best 

 observed when the tube contains either only one stratum, or a small 

 number separated by a broad interval. There is reason to believe that 

 even in the dark discharge there may be strata ; for we have found a 

 development of heat in the middle of a tube, in which there was no 

 illumination except on the terminals. 



8. Even when the strata are to all appearance perfectly steady, a pul- 

 sation can he detected in the current ; hut it is not proved that the strata 

 depend upon intermittence. 



9. There is no current from a hattery through a tube divided hy a glass 

 division into two chambers, and the tube can only be illuminated hy 

 alternating charges. 



10. In the same tube and with the same gas, a very great variety of 

 phenomena can be produced hy varying the pressure and the current. 

 The luminosities and strata, in their various forms, can be reproduced in 

 the same tube, or in others having similar dimensions. 



11. At the same pressure and with the same current, the diameter of 

 the tube affects the character and closeness of the stratification. 



II. "Note on Legendre's Coefficients." By I. Todhunter, 

 F.RS. Received April 16, 1878. 



In the "Proceedings of the Royal Society," vol. xxvii, pp. 63-71, 

 Professor Adams has given, by an inductive process, the development 

 of the product of any two of Legendre's Coefficients in a series of the 

 Coefficients; from this is immediately deduced the value of the in- 

 tegral between the limits —1 and +1 of the product of any three of 

 the Coefficients. On the other hand, if we know the value of this 

 definite integral, we can immediately deduce the development of the 

 product of any two of the Coefficients. Thus it may be of interest to 

 give a brief investigation of the value of the definite integral. I 

 follow the notation adopted by Professor Adams. 



The formula to be established is 



f l p tj-pj .- 2 A(g-wQA(s-tt)A(s-j?) 

 J-i m n P ^~2s-+T AW ' 



where 2$=m+n+p, and the functional symbol A(r) is thus defined: 

 if r is a positive integer 



2 c 2 



