He was one of the early founders of the theory of determinants; in particular, he invented the functional determinant formed of the n 2 differential coefficients of n given functions of n independent variables, which now bears his name (Jacobian), and which has played an important part in many analytical investigations (see Algebraic Forms).

Phys., 1905, 18, p. 941) founded his theory of aberrations on the differential geometry of surfaces.

The analytic method sought to express the moon's motion by integrating the differential equations of the dynamical theory.

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A belted arrangement of relief s and soils, resulting from differential erosion on strata of unlike composition and resistance, characterizes almost the entire area of the coastal plain.

Where accurate differential observations or photographs involving other than instantaneous exposures have to be made, the additional condition is required that the optical axis of the telescope shall accurately and automatically follow the object under observation in spite of the apparent diurnal motion of the heavens, or in some cases even of the apparent motion of the object relative to neighbouring fixed stars.

Everywhere are evidences of water and wind erosion, of desiccation and differential weathering.

His treatises and contributions to scientific journals (to the number of 789) contain investigations on the theory of series (where he developed with perspicuous skill the notion of convergency), on the theory of numbers and complex quantities, the theory of groups and substitutions, the theory of functions, differential equations and determinants.

The other method starts from the observed values of the periods, and establishes a differential equation from which these periods may be derived.

He also showed that every equation of an even degree must have at least one real quadratic factor, reduced the solution of linear differential equations to definite integrals, and furnished an elegant method by which the linear partial differential equation of the second order might be solved.

And the more advanced Imperialists, as well as the more oldfashioned protectionists (like Mr Chaplin) who formed an integral body of the Conservative party, had looked forward to this tax being converted into a differential one between foreign and colonial corn, so as to introduce a scheme of colonial preference and commercial consolidation between the colonies and the mother country.