392 Mr Baker, On Abelian Functions, &c. [Jan. 24, 



e notation 



Hence, retaining the notation for (/> (£), <$(£), if 



A^B 



so that 



A£+B = a'[a (r) £+ £«] + /3' [7"'^+ S«], 

 we have 



when e r = 1, vie,. + 5 = a'c/ + /3' = - (ac,. + #), 



when e,. = — 1, Aa r + B = a' a/ + /3' = — (aa r + /3); 



we find also, in fact, in these two cases, 



<f> (c r ) = — cf>'%. (c r ), <£ (a r ) = — cfi'%. (a r ). 



Thus it follows, that when e,. = 1, the series 



M£0)-2, -i , 



m bm m "nib ~ r J »i 



vanishes for £=c,. ; for when £= c r the term arising by the 

 substitution 0, of the group, is equal and opposite to the term 

 arising by the substitution $%.. Similarly when e r = — 1, the 

 series vanishes for £=«,.. 



In a somewhat artificial way this result may be included in 

 the statement — the series, \ (£, a), has, beside £= 00 , the p zeros 



Oj\ , Cj tti — Ci Qlp -p Cj5 dp — Cjt 



?; £i ^ ..... 



C ^ 2 ' 



where e l3 ..., e p , each equal to + 1, are the signs determining the 

 p fundamental substitutions, as given by putting n = 1 in the 

 formulas written down above for the substitution ^! . 



(3) On the production of a cloud by the action of ultra-violet 

 light on moist air. By Mr C. T. K Wilson, Clerk Maxwell 

 Student. 



If the light from an arc lamp be brought to a focus, by means 

 of a quartz lens, within a vessel containing moist dust-free air, a 

 bluish fog becomes visible in the course of a few minutes along 

 the path of the light. The cloud particles remain in suspension 

 for hours after the light has been cut off. The phenomenon is 

 shewn even in unsaturated air, but the faint blue haze which 

 then developes takes much longer to form. When the radiation 

 is not sufficiently intense to shew these effects a dense fog can 

 still be obtained by bringing about slight supersaturation by 

 expansion. These clouds, unlike those obtained by Tyndall (Phil. 



