1874.] On the Meteorological Use of a Planimeter. 435 



The simple functions or algorithms whose properties and relations it 



z 



is the province of this new calculus to determine are 



I. (1 7?,) A, sin z sin 2, and their inverse functions, 

 -h +h 



z n'*'^ z ( or / s )t 1S z -> a logarithm taken to the base (1-f h) h , or 



(I Ji\~k an( l sin- 1 2 sin- 1 z. 

 ^ '^.^, 



The calculus so founded the author proposes to call the Calculus of 

 Factorials. 



The branches of the subject treated of in the present memoir will be 

 understood from the following list of the contents of the various sections 

 into which it is divided : 



Ch. I. 1. Definition and properties of z n /*^ z , or more generally 

 TA2r , when n is a whole positive number. 



2. Factorials with a negative whole index. 



3. Factorials of which the index is a positive fraction. 



4. Factorials of which the index is a negative fraction. 



5. Factorial radicals. 



Ch. II. 1. Application of the theory of finite differences to fac- 

 torials. 



2. Differenciation * of factorial exponentials and factorial lo- 

 garithms. 



3. Development of the various simple functions into factorial series. 



XVII. "On the Employment of a Planimeter to obtain Mean 

 Values from the traces of continuously Self-recording Meteoro- 

 logical Instruments." By ROBERT H. SCOTT, M.A., F.R.S. 

 Received May 23, 1874. 



It is hardly necessary to remind the Fellows that the self-recording 

 instruments employed by the Meteorological Committee at their Obser- 

 vatories for the continuous registration of pressure and temperature 

 furnish their results in the form of photographic traces. The usual 

 method of dealing with these barograms and thermograms, as they are 

 respectively called, is to measure them, at certain intervals by appropriate 

 scales, and to treat the numerical values so obtained by arithmetical 

 processes so as to arrive at mean results. 



This method is naturally very laborious, and its accuracy is to some 



* The author uses this word to denote that which in the calculus of finite differences 

 takes the place of differentiation in the differential calculus. 



