68 
the five ratios of the six indeterminates A, a, j3, . . Z and 
wherein X is a function of x only which vanishes when the 
integration with respect to v is definite. If we put A - a 
then (1) becomes a linear differential equation of the first 
order and we have still four disposable ratios. 
Oakival , near Brisbane , Queensland , 
Australia, October 17, 1867. 
“ On some Points in the Restoration of Euclid’s Porisms,” 
by T. T. Wilkinson, F.R.A.S., Corresponding Member of 
the Society. 
M. Chasles, in his Apergu Historique, p. 282, says that : 
“ Diophante dans ses Questiones Arithmetiques, a plusieurs 
fois employe le mot Porisme,* pour designer certain propo- 
sitions concernant la th^orie des nombres, sur lesquelles il 
appuie ses demonstrations, et qui formaient probablement 
un ouvrage qui ne nous est pas parvenu. Voir, par exem- 
ple, les Propositions 3, 5, et 19 du Livre V.” This very 
valuable work was published at Brussels in 1837. The 
same Author in his restoration of Les Trois Limes de 
Porismes D’Euclide, published at Paris in 1860, gives an 
analysis of these Porisms in pp. 47 — 51, and adds to their 
Le Porisme est une proposition dans laquelle on enonce une verite, en 
affirmant qu’on peut toujours trouyer certaines choses qui la completent.” 
— (Chasles, Porismes p. 54.) 
This new definition regards Porisms as incomplete Theorems. — M. Breton 
( Porismes , 1866), regards them ratheras incomplete Problems. 
“ A Porism proposes to demonstrate that one or more things may be found, 
between which and innumerable other objects assumed after some given law 
a certain specified relation is to be shewn to exist.”— (Leslie, Geom. Anal.') 
“A Porism is a proposition affirming the possibility of finding such con- 
ditions as will render a certain problem indeterminate, or capable of innumer- 
able solutions.” — (Playfair, JEdin. Trans.) 
“ Porisma est propositi© in qua proponitur demonstrare rem aliquam, vel 
plures datas esse, cui, vel quibus, ut et cuilibet ex rebus innumeris, non 
quidem datis, sea quse ad ea qua? data sunt eandem habent rationem, convenire 
ostendendum est affectionem quandum communem in propositione descrip- 
tam.” — (Simson, Opera.) 
“ Porisma vero, quod proponitur in pofismum, hoc est in inventionem et 
investigationem propositi.” — (Commandine, Pappus , Vol. II, p. 160.) 
