228 Royal Society :— 
laws of appension are shown to be the same as those of addition, 
and are hence expressible by the same signs of combination, the 
difference in the objects combined preventing any ambiguity. We 
thus get the conception of a point as an annihilated line. 
The tensor operation, considered dynamically, leads to the opera- 
tion of changing a line dynamically so that it should bear the same 
relation to the result as two given lines bear to each other in magni- 
tude and direction. This assumes three principal forms according 
to the difference of direction. If there is no difference of direction, 
the operation is purely a tensor. If the directions differ by a semi- 
revolution, the rotation of one line into the position of the other may 
take place on any plane. The operation is then termed a negative 
scalar; the tensor, which includes the operation of turning through 
any number of revolutions, is distinguished as a positive scalar, If 
the rotation be through any angle, but always on the same plane, 
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 and 
negative exponents, and the binomial theorem for these powers. It 
also induces us to consider the laws of formators, 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 
