134 Sir James Cockle on Primary Forms. 



cinders and burning matter, together with flames, rumbling, and 

 earthquakes. These phenomena, which had a gradually decreas- 

 ing course, lasted through all the month of July 1874 ; and traces 

 of them still remain. 



Stromboli also in June last had an unusual eruption, being 

 violently agitated, and throwing out stones as far as the inha- 

 bited district which lies underneath it, showing much greater 

 activity than in the little explosions every two or three minutes 

 which characterize its usual action. It seems that Vesuvius also 

 has not remained indifferent to all this ; and I saw myself from 

 its crater, as well as from that of Stromboli, a remarkable and 

 unusual amount of thick smoke coming out at the same time 

 that the eruption of Etna took place. 



XVII. On Primary Forms. By Sir James Cockle, F.R.S., 

 Corresponding Member of the Literary and Philosophical Society 

 of Manchester, President of the Queensland Philosophical So- 

 ciety, <Sfc* 



I. {COMPARING p. 428 of Boole's 'Differential Equations' 

 V7 (1865), and pp. 184 and 190 of the Supplement, with 

 a citation in Mr. Harley's recent paper " On the Theory of Dif- 

 ferential Resolvents'" (which purports to be reprinted from the 

 Report of the British Association for 1873), it seems to me that 

 Boole did not always use the term primary in the same sense. 

 By primary I mean integrable, but not through Boole's reduc- 

 tions. By a factorial substitution I mean a change of y into Xy, 

 where X is a function of x. By taking the criticoid of a biordinal 

 I mean reducing its middle term to zero and dividing the equa- 

 tion so transformed by the coefficient of its first term. This 

 criticoidalt transformation is effected by factorial substitution, 

 not by change of the independent variable. By the equation of 

 the caesura, or briefly the casura, I mean an equation derived 

 from a given equation by expunging its last term and diminish- 

 ing by unity the indices of the differential coefficients. 



2. The regular forms, as we may term those solved through 

 Boole's reductions, are of two descriptions. In one factors are 

 preserved which in the other are lost. The first description of 

 binomial biordinal may be written 



{(A-2m)(A-l)-(v + 2n)(v-l)* 9 }y = 0, . (1) 

 where 



A = D+/3 and y = D-f a. 



* Communicated by the Rev. Robert Harley, F.R.S. 



t As to this user of the term criticoidal, see my paper " On Hvperdis- 

 tributives " (Phil. Mag. for April 18/2) and the papers connected there- 

 with. 



