322 



REPORT 1875. 



Yalores numeri n. 



viz. suppose the numbers in the second 4-line known : then simply moving 

 these each five steps onward we have the (auxiUary) numbers of the first 

 5-lin.e ; and thence by a mere addition the required series of numbers shown 

 by the second 5-linc. And similarly from this is obtained the second 6-line, 

 and so on. 



35. More extensive tables are contained in the memoir 



Marsano, " Sulle legge deUe derivate generale delle funzione di funzione 

 et Bulla teoria delle forme di partizione dei numeri intieri." (4°, Geneva, 1870), 

 pp. 1-281 ; and three tables paged separately, described merely as " Tavoli 

 dei numeri C, „ S,,?, ^'q, e citate nel teste coUe indicazioni di Tavole I., II., III., 

 ai n' 77, 79, 81;" viz. the reader is referred to these articles for the ex- 

 planations of what the tabulated functions are ; and thei'e is not even then 

 any explicit statement, but the investigation itself has to be studied to make 

 out what the tables are. It is, in fact, easier to make this out from the 

 tables themselves ; the explanation is as follows : — 



Table I. (16 pages) is, in fact, Euler's table, showing in how many ways 

 the number n can be made up with the parts 1, 2, 3 . . . «i ; but the extent 

 is greater, viz. n is from 1 to 103, and m from 1 to 102. The auxiliary 

 numbers given in Euler's table are omitted, as also certain numbers which 

 occur in each successive line ; thus a specimen is 



n= 1 2 3 4 5 6 7 8 9 10 &c. 



Co, n 

 Ci, „ 



C4,» 



&c. 



where the line C^, „ (ways of making up n with the parts 1, 2, 3, 4) is 1, 1, 

 2, 3, 5, 6, 9, 11, 15, 18, &c., viz. we read from the corner diagonally down- 

 wards as far as the 5, and then horizontally along the Uue : this saves a 

 large number of figures. The table is printed in ordinaiy quarto pages, 

 which are taken to come in in tiers of seven, five, and three pages one under 

 the other, as shown by a prefixed diagram; and the necessity of a large 

 folding plate is thus avoided. 



The successive lines give, in fact, the coefficients in the expansions of 



1 1 1 1 



l_.r' 1_A'. l-.r^' 1-.1-. l-.r' . 1-x'" " " l-.v. l-x'. ... 1-^ 

 each expanded as far as .r"'. 



102' 



