©24 REPORT — 1895. 



11. We may now make our variables .r, , . . .r„, the time differentials of another 

 set I, . . . ^n, that is, 



And/(rti . . . c^da^ . . . <Za„ is now the chance that when i = 



dt ' 



flj + «?«], 



-^ = a^ . . . a^ + da^, &c., 



and we now use for our linear functions 



X,ai+Xo(«i + Sffi)+ . . . =5<2^S 



the summation including all the successive values of --^l at the successive times 

 i = 0, t-ht, t = 28t, &c. Similarly, for the second linear function we take 



hll ^2 and so on. 



dt 



12. Then we may make ht infinitely small and N infinitely great, while l&ht = T 

 and our linear functions become 





or ^i-Xi, I2-X2, &c., if Xj, X2, &c., now denote the initial values of ^j, l^, &c. 

 And our proposition now assumes the form that the chance of |j . . , ^„ lying 

 within assigned limits varies as 



'I|(«,(li-X,)^ + ^.(^:-X)(^,-X,), &C.) }, 



where the index is a quadratic function of (^1 - X,), (^,_, - X,,), &c. 



We thus get the exponential form, or ' law of error,' in all cases for which the 

 modified condition of independence is satisfied. In order that it may be satisfied 

 the time variation of I, . . . ^„ must depend not only on the values of k\ • • • In for 

 the time being, but also, in some appreciable degree, on external fortuitous causes. 

 And the greater the comparative importance of these fortuitous causes, the shorter 

 is the time T which, as the interval between two successive states of the system, 

 makes the variables in one of those states independent of those in the other. 



Department II. Meteorology. 



1. Prohable Projection Lightning Flashes. 

 By Eric Stuart Bruce, M.A. Oxon., F.R.Met.Soc. 



■ The classification of lightning flashes is already beset with diflSculties. The 

 object of this paper is to suggest the possibility of Projection Lightning Flashes, 

 the existence of which would increase the difficulties of classification. 



Sheet lightning refiected on clouds is made up of numerous images of the 

 lio-htning flash superimposed one upon the other. If there is a cloud with a suffi- 

 ciently small opening in it between the lightning discharge and a reflecting surface 

 of clouds, the latter being in an adequate position, on the clouds that otherwise 

 would have been merely illun^iiiated with sheet lightning will appear the optical 

 proiection of a lightning flash. If the surface of clouds upon wiiich the image 

 falls is level, the image will be a perfect reproduction of the lightning flash, save 



