90 



the operation is here termed a clinant. If the rotation may take 

 place on any variable plane, the operation is a quaternion. 



The laws of scalars are immediately proved to be the same as 

 those of tensors, but in addition they introduce the idea of negativity. 

 This enables us in the algebra of scalars, to invert a sum generally, 

 and thus allows of a perfect inversion of the first two formations. 

 But a power with a fixed integral exponent can only be inverted on 

 certain conditions. This partial inversion, however, leads to a solu- 

 tion of quadratic equations, and to a proof that formations consisting 

 of a sum of integral powers, cannot be reduced to null by more 

 scalar values of the variable than are marked by its highest exponent. 

 Hence if such a formation is always equal to null, all the coefficients 

 of the variable must be null. We thus obtain the method of inde- 

 terminate coefficients, by which we are enabled to discover a series 

 which obeys the laws of repetition with respect to its variable, and 

 becomes equal to a power when its variable is an integer. This 

 enables us to define a power with any index, as this series, and hence 

 to attempt the inversion of powers with variable indices, which we 

 succeed in accomplishing under certain conditions. This investigation 

 introduces the logarithm of a tensor, powers with fractional arid 

 negative exponents, and the binomial theorem for these powers. It 

 also induces us to consider the laws offormators, or the operations 

 by which a formation of any variable is constructed. They are 

 shown to be commutative and associative in addition, associative in 

 multiplication, directly distributive and repetitive, but not generally 

 commutative in multiplication, nor even inversely distributive. "When 

 formators are commutative in multiplication and distribution, they 

 are entirely homonomous with scalars, which may even be considered 

 as a species of formators. The results of the former investigation, 

 therefore, show that logarithms, fractional and negative powers, and 

 the binomial theorem hold for these commutative formators. 



The necessity of tabulating logarithms and of approximating to 

 the solutions of equations, leads to the consideration of a method of 

 deriving consecutive values of formations for known differences of 

 the variable, and of interpolating values of the same formation for 

 intermediate values of the variable ; that is, the algebra of differences. 

 Considering the two operations of altering a formation by increasing 

 the variable, and taking the difference between two different values 



