TRANSACTIONS OF THE SECTIONS, 7 
of the solid; P the force required to overcome the friction; then 
4 ase 
poly Lz (144sin? B-+sin4 8). 
In ordinary cases, the value of f for water sliding over painted iron is 0036. The 
quantity Is z (144 sin? 6+ sin‘ 8) is what has been called the “ augmented surface,” 
In practice, sin‘ 6 may in general be neglected, being so small as to be unimportant. 
Some Account of Recent Discoveries made in the Calculus of Symbols. 
By W. H. L. Rosser, A.B. 
Before the publication of Professor Boole’s memoir on a “General Method in 
Analysis,” which appeared in the ‘Philosophical Transactions’ for 1844, those 
mathematicians who adopted the symbolical methods suggested by the researches 
of Lagrange and Laplace, confined themselves to the use of commutative symbols, 
and the science was consequently very limited in its applications. It received a 
fresh impulse from the very remarkable memoir of Professor Boole mentioned above, 
in which an algebra of non-commutative symbols was invented and applied to the 
integration of a large class of linear differential equations. It occurred to the author 
that the proper method of extending the calculus was to construct systems of 
multiplication and division for functions of non-commutative symbols. This he 
Pitingly effected in his memoir published in the ‘Philosophical Transactions’ 
for 1861. “As the symbols are non-commutative, two distinct systems of multi- 
— and division, internal and external, arise for each class of symbols em- 
oyed. 
q Let p and r be two symbols combining according to the law 
F (n). pm=pmf (x-+m), 
where f (7) is any function of (m), then he gave, in the memoir alluded to, equa- 
tions to determine the conditions that a symbolical function such as 
pr dy (m) +p" bya (m) +P” bn—a (a) + &e. +h (m7) 
may be divisible internally and externally without a remainder by the symbolical 
function py, (7)+, (7), where 
Pn (7) Pr—r (7); Pn—a (F) +++ Po (m7), Pr (w) and y, (7) 
are all rational functions of (z), or, in other words, that py, (+) +, (7) may be an 
internal or external factor of p* (m)+p"—' hy _1(7)+ &e., and also an equa- 
tion to determine the condition that y, (p) «7+ (p) may be an internal factor of 
h: (p) +m +h. (p) m+, (p) «m+ (p): 
He then gave some theorems for the transformation of certain functions of these 
symbols, which lead to some very curious theorems in successive differentiation : he 
has treated this part of the subject more fully in the ‘ Philosophical Magazine’ 
for April 1862. In a subsequent part of his paper in the ‘Philosophical Transac- 
tions,’ he established binomial and multinomial theorems for these symbols, by 
showing how to expand 
(p?+ pO ())” and (p%+p*—? 8, (7)+p*? 6, (7) + ....)” in terms of (p) and (7). 
At the end of the paper he gave some methods for solving differential equations 
by a process analogous to the “Method of Divisors” in the theory of algebraical 
equations. In his second memoir “On the Calculus of Symbols,” published in 
the ‘ Philosophical Transactions’ for 1862, he has shown how we may find the 
highest common internal divisor of functions of non-commutative symbols, and 
also how we may resolve them in all possible cases into two equal factors, a process 
analogous to that for extracting the square root in common algebra. He then in- 
vestigated the theory of multiplication in this calculus more generally. He gave a 
rule to find the symbolical coefficient of p™ in a continued product of the form 
(p+, ()) (p45 (m)) (p+4, (7) vvsseees (P4On (H)): 
After this he resumed the consideration of the binomial and multinomial theorems 
explained in the former memoir, He gave the numerical calculation of the coefti-~ 
