28 Observations on the Notations employed in 



volume of the Cambridge Transactions, printed in 1827, con- 

 tained the first specimen of the index subscript-notation, though 

 he says Prony and others had previously employed it; the 

 subscript notation, or the calculus of differential coefficients, 

 has found its way into some treatises on mathematical subjects. 

 It is supposed the above examples will be sufficient to indicate 

 the difference between the subscript notation and that more 

 generally used ; but in order to become a proper judge of the 

 difficulty in reading a book written in the new notation, after 

 one has been accustomed to the common one, the reader must 

 go through the task himself; and his qualification to give an 

 opinion will be all the better, if he have to commence with 

 finding out what the new notation really means. 



The notation of Lacroix (that is the notation employed by 

 him) has been so generally used in mathematical works du- 

 ring many years, that some strong reason ought to be given for 

 introducing another ; on this head, however, I have met with 

 only one advocate, namely, the author of a small work entitled 

 " On the Notation of the Differential Calculus." The work 

 is said to be scarce ; my copy has no title-page, but the book 

 was printed by Metcalf, Cambridge, some time ago. The 

 author is understood to be a very distinguished member of 

 the University ; the reader should refer to the book for prac- 

 tical illustrations and for the full scope of the writer's object; 

 only some extracts, strictly bearing upon the point under 

 consideration, can be taken on the present occasion. 



Art. 32. The author says, " We must observe that since 

 d u is obtained from u by performing upon it some operation 

 with regard to .r, of which it is a function, it is necessary 

 when u is a function of several independent variables x, y, z, 

 ... to know with regard to which of them the operation d is 

 to be performed, for the results may be very different accord- 

 ing as d is performed with regard to the one or the other. 

 And herein the notation of differentials is defective, for when 

 we meet with the expression d u it is impossible to know what 

 it represents. We know, indeed, that an operation of a cer- 

 tain kind is to be performed on u in respect to some quantity 

 of which u is a function, but which is that quantity it is im- 

 possible to tell. Hence arises all that confusion and obscurity 

 from which very few, if any, treatises on differentials are free ; 

 and in this respect also it is very much inferior to the cal- 

 culus of differential coefficients, which is remarkable for its 

 perspicuity." 



Art. 4-2. " Whenever u is a function of one independent 

 variable, the differential coefficients may be represented by 



„ . du d?u d 3 u 

 the fractions -^ , ^, j& 



