1876.] On Clairautian Functions and Equations . 43 



northern stations, yet, to attain the maximum shown at Sl'^ 12'', the 

 mercury lose 0*4:0 inch within twelve hours during the 31st of March. 

 I need scarcely point out the weighty bearing which these facts must 

 have on all investigations with reference to the great barometric oscil- 

 lations within our latitudes as well as to those of lesser magnitude within 

 the tropics. 



The following Table contains the daily mean height of the barometer 

 at each station at the hours of maximum and minimum previously given, 

 together with the mean height for the year. 



Stations. 



1st Max. 



Min. 



2nd Max. 



Mean of 

 year. 





in. 



in. 



in. 



in. 





80-037 



29-670 



29-816 



29-794 



Pekin 



30131 



29-701 



29-809 ? 



30015 





30155 



28-848 



30142 



30-058 





28-309 



28-239 



28-289 ? 



28-296 



Makerstoun 



29-959 



29-770 



29-915 



29-586 



Singapore 



29-947 



29-864 



29-918 



29-895 



Madras 



29-854 ? 



29-785 



29-844 



29-853 





23-184 



23-118 



23-221 



23-195 



Catherinenburg 



28-967 



28-624 



29-099 



29-023 



Bogoslowsk 



28-567 



28-298 



28-708 



28-746 



It will be perceived that the minimum height was less at all the stations, 

 with the exception of Makerstoun, than the mean for the year. 



VI. On Clairautian Functions and Equations. By Capt. Allan 

 Cunningham, Hon. Fellow of King's Coll. Lond. 



(Eoorkee, India) . Communicated by Prof. Cayley. Received 

 April 18, 1876. 



(Abstract.) 



Notation. — In this paper D stands for y\ y^' y^'^^ stand for the diflFerential 



coefficients of y (and therefore y^ is equivalent to y itself) ; X, Xj, Xg, &c. stand for 

 known functions of x; X', X" X('«^, X/, X," X/*"), &c. stand for the dif- 

 ferential coefficients of X, X^^, &c, ; y^ stands for a particular integral of a linear 

 differential equation ; y^ stands for the complete arbitrary portion of the solution of a 

 linear differential equation. 



1- Clairautian Functions. — It is proposed to apply the term Claikau- 

 TiAN rusrcTiON to the following expressions (which possess properties 

 similar to that on which the solution of " Clairaut's equation " is 

 founded), viz. 



yO'\ kyO.-'> - ^f", + . 1^2,<.-. + MV'"> • (1) 



and to denote them by the symbols ^IJo, „, ^IJi^ „, ... .^"U,,, „, so that 



