94 Prof. Challis on a particular case of the 



dental causes, the average aspect nevertheless indicates the 

 law of change alluded to, seeing that there appears to have 

 been an aera in which the earth was occupied exclusively by 

 cold-blooded creatures requiring but little oxygen ; that it was 

 subsequently inhabited by animals of superior organization 

 consuming more oxygen, and that there has since been a con- 

 tinual increase of the hot-blooded tribes and an apparent di- 

 minution of the cold-blooded ones. 



Bearing in mind therefore the undoubted relationship that 

 exists between the consumption of oxygen on the one hand 

 and the degree of vitality and height of organization on the 

 other, it would appear extremely probable that there is some 

 connexion between the supposed change in the vital medium, 

 and the increased intensity of life and superiority of construc- 

 tion that has accompanied it. Whether the alteration that 

 has taken place in the constitution of the atmosphere is to be 

 looked upon as the cause of this gradual development of or- 

 ganic existence, or whether it is to be regarded as an arrange- 

 ment intended to prepare the earth for the reception of more 

 perfect creatures, are points which need not now be entered 

 upon. The question at present to be determined is, whether 

 the alleged improvement in the composition of the air has 

 really happened, and if so, whether that improvement has had 

 anything to do with the changes that have taken place in the 

 characteristics of the earth's inhabitants. 



[*** We have inserted the foregoing communication because the inquiry 

 to which it relates is one of extreme interest, and our correspondent's rea- 

 soning may excite attention to the subject; but we doubt whether his 

 inferences are warranted by the known facts of Chemistry and Palaeon- 

 tology : we fear that his generalizations are at least premature. — Edit.] 



XVI. On a particular case of the Application of Criteria of 

 Integr ability. By the Rev. J. Challis, M.A., Plumian 

 Professor of Astronomy in the University of Cambridge*. 



TITHENEVER the differential expression P d x 4- Qdy 



* " is the differential of some function (u) of the variables 



x and y, P and Q are respectively equal to the partial differ- 



. , ^ . du . du TT dP d* u , 



ential coefficients -=— and - 1 —. Hence -*— = -3 — 5—, and 

 d x dy dy dy d x 



d Q d 2 u XT . d* u , d*u . . . , 



-r? =a -. ^-. Now since -. — t— alK » 1 — -7- are identical 



d x dx dy dy dx d x dy 



quantities, the order of differentiation being of no consequence, 



it follows that -r— and *s — are also identical. This is the sole 

 dy d x 



* Communicated by the Author. 



