We define
$$\varphi _ { p + 1 } \left( v , v ^ { \prime } \right) = { } ^ { t } V \Delta _ { p + 1 } V ^ { \prime } = v _ { 1 } v _ { 1 } ^ { \prime } - \sum _ { i = 2 } ^ { p + 1 } v _ { i } v _ { i } ^ { \prime }$$
and
$$q _ { p + 1 } ( v ) = \varphi _ { p + 1 } ( v , v )$$
Express $\varphi _ { p + 1 } \left( v , v ^ { \prime } \right)$ as a function of $q _ { p + 1 } \left( v + v ^ { \prime } \right)$ and $q _ { p + 1 } \left( v - v ^ { \prime } \right)$.