Transposition and ApP
Analogous to the SE, the 
also holds under transposition, that is,
and after transposition
A scalar measure of refractive power is frequently used to evaluated refractive outcomes. Such scalar measures are based on the average power of a lens or lens system, for example, spherocylinder.
A commonly used measure has been the mean spherical equivalent usually referred to as the spherical equivalent (SE). Harris recognised, however, that there may be other scalar terms for the SE.1 As such, it has recently been shown that the 
which in addition to orthogonal rays, includes the average or mean of oblique paraxial rays. It is also associated with better visual acuity than the SE and therefore may be of more clinical use.2 The relationship between the 
and SE, however, needs clarification.
There are currently two methods used to calculate the average paraxial power of a lens:
SE, which is derived from the average orthogonal paraxial power3
.Average paraxial power
, which is derived from the average orthogonal and oblique paraxial powers2 
Consider the following lens system F written as an optical cross where C1 and C2 are two orthogonal lens cylinders at axes a and a±90:
This can be separated into the sum of two paraxial lens systems:
often then written in sphere/cylinder x-axis ( 
) form as 
, based on the assumption that two equal orthogonal cylinders equal a spherical lens, that is, 
The average power 
of the system is the sum of the average power of each component, that is, 
where 
, 
and 
are the average paraxial powers of each of the three lenses.
For the SE, the average is the average of orthogonal paraxial rays, that is,
SE=average orthogonal paraxial power 
So that
While for the 
, the average is the average of orthogonal and oblique paraxial rays, that is,
=average orthogonal and oblique paraxial power 
Therefore, the 
is equal to half the SE if calculated in cross cylinder form, that is, 
Transformation
Transforming an optical cross of cylinders into a spherocylinder 
often rests on the assumption that two equal orthogonal cylinders equal a spherical lens,3 that is, 
or 
. This is an approximation, as it has been shown that two equal orthogonal cylinders are not equal to a spherical lens3 and in addition, the intersection of two equal orthogonal cylinders is Steinmetz solid rather than a sphere.2 It would, therefore, be more accurate to state that within a paraxial ray system that two equal orthogonal cylinders only approximate a spherical lens.
If 
, where 
represents the resultant spherical lens, 
; and 
where 
, represents the resultant spherical lens 
.
Then, based on this approximation, the lens system is written in 
notation as:
It is important to note, however, that the derivation of the SE is based on treating a lens cylinder as a cylinder and not on the assumption that two orthogonal cylinders equate to a spherical lens.
Although not strictly correct, the SE is often calculated as:
where 
and 
It is also important to note that the error introduced by this approximation is not constant and the magnitude of the error depends on the power of the two lenses.3 Therefore, application of the formula for the SE based on the assumption that two orthogonal lenses equate to a spherical lens should be limited to small powers.
If a refractive error, however, does comprise a true spherical component, then the average of that component is of course the sphere.
Therefore, for a refractive power that contains a sphere, the 
and the 
and the difference between the 
and SE is equal to 
.
Conclusion
While any scalar measure of refractive error loses dimensions of sensitivity, scalar measures of power remain important in many applications. The 
as a scalar measure is more inclusive and appears to be associated with better visual acuity than the SE, although further clinical trials are needed particularly in different age groups and conditions. It will also be important to explore and evaluate the 
with higher refractive errors. If the evidence remains supportive, then useful clinical applications of the 
would include, for example, providing an equivalent lens for a person who is unable to tolerate the toric prescription as this is associated with less degradation of visual acuity than may occur with the SE.
