I read the paper you linked and, actually, Pareto doesn't specify what equations is referring to. But I think he speaks of equilibrium between demand and supply in a market. So, I suppose he is referring to equilibrium in classical mechanics, and to equations as Lagrange or Newton equations in mechanics.
Pareto, I think, is comparing the equilibrium of mechanical objects with the equilibrium in a market when demand equals supply. The ‘forces’ that make prices change, that is excess demand or supply, are represented by the dynamic equations that rule the movement of prices. Such equations may be viewed as an analogous of equations of mechanics.
I say that because Pareto explicitly speaks of demand and supply equilibrium and of an analogy with the equations of classical mechanics in his Principles of pure political economy.
For instance, he speaks of certain equations he had written before, relative to demand/supply and prices and says that
In the study of economic equiibrium [such equations] have an analogue function as the equations of Lagrange in the study of equilibrium of mechanics. (Corso di economia politica, p. 152, italian edition, UTET, 1987, my translation from italian).
Moreover, Pareto speaks of mathematics in economics, comparing it with classical mechanics, in his Applied political Economy: here we can see that he knows well classical mechanics.
As far as I know, Applied political Economy hasn't been tranlated in english.
But I speak Italian, so I can read Pareto’s operas in italian.
I read some passages of his 'Applied Political Economy’, and I can translate some here.
In this work Pareto devotes some pages to a comparison between economic system and mechanical systems and defends the application of mathematics to economics, particularly mathematical tools analogous to the classical mechanics equations.
In paragraph n. 592 of Applied Political Economy, Pareto makes an explicit comparison between economics concepts and mechanical concepts.
He devotes a whole page to a scheme, where he puts on the left of the page mechanical concepts and on the right economics concepts.
He writes (ibid., p. 646):
The equilibrium of an economic system presents surprising analogies with the equilibrium of a mechanical system. When one knows well this latter equilibrium, has also clear ideas about the former.
Then he proceedes with the scheme, comparing 'mechanical phenomena' with 'economics phenomena'. For example:
Given a certain number of material bodies, the relations of equilibrium and motion are studied[...], abstracting from other properties.
Given a society, the relations are studied which production and exchange arouse among people, abstracting from other circunstances.
And he continues the scheme with other similar analogies.